455 
Mr. Herschel on various points of Analysis. 
0^^ . Cos X0 
zn 
where 
Y __ (—0''^. (o2«— 2^’— I 
2X 
-0 
zn — 2x 
V. 
B 
■I . 1.2 
• • • 
zn—zx 
I ( 1 v/ ~i) 
{zn — 2.r) 
1 ^ • 1.2. . . {2X) I 
Which compared with tlie value of before found (so) 
gives the following singular equations, 
I r 
(23) 
I “2,t— 2 . «2,v— 4 . “o-ll/—, 
+ -TT- + TyiT- + • • . . — - 
1.2 
1 .2...(2JC) 
'^2X= 1 . + } — 
L I ' 1.2.3 1.2 .. . (20; 1 J 
2 * i. 2 ..( 2 .r) ‘ (^4) 
The latter of these two equations affords the even values of 
in terms of the odd, and hence we are enabled to e.xpress the 
sum of the series 
I 
1 1 &C 
2X-j- I 
C(i)' 
3" ' ■ S 
by means of the numbers of Bernouilli, which Euler appears 
to have considered as impracticable.-f We need hardly re- 
* See “ Essay on Logarithmic Transcendents,” page 51. I should not omit to 
observe that the equations in p. 69 of that work, exjiressing the value of the function 
”C (a:) + (—1)” . ”C (jT when combined with our equation (10) by making 
t 
F {t) {t) — — h &c, afford a series of results, highly interesting, but 
which the necessary limits of this paper forbid me at present to dilate on. 
f “ Per hos autem numeros Bernouillianos secans e.rprimi non potest, sed requirit 
alios mimeros qui in summas potestatum reciprocarum imparium ingrediuntur, si 
** enim ponatur, 
III. r 
1 ““ — ~~ ““ — “ -p &C __ oc, • — — 
3 5 7 2 * 
I — — — ^ — — — “T “F 
3 * 5 7* J-2 2 
3 N a 
